آشوب و بررسی آن در سیستم های بیولوژیکی

ارائهراحله داودی

استاددكتر فرزاد توحيدخواه

1388دی

دانشگاه صنعتي اميركبير

دانشكده مهندسي پزشكي

What is talked in this seminar:Introduction to chaosChaos propertiesHistory FractalsChaos and stochastic processLogistic Map

What is talked in this seminar: (continue)Biological models producing chaos Chaos in heart sign of healthy or disease?Application:

Model of heart rate Applying chaos theory to a cardiac

arrhythmia

What chaos is:

One behavior of nonlinear dynamic systems

Unpredictable for long time but limited to a

specific area (attractor)

Seems to be random while it happens in

deterministic systems

Highly sensitive to initial condition

Chaos Properties:

• Fractal (Self Similarity)

• Liapunove Exponent (Divergence)

• Universality

Henri Poincaré - 1890while studying the three-body problem, he found that there can be orbits which are non-periodic, and yet not forever increasing nor approaching a fixed point.

History

Poincare &Three body problem

The problem is to determine the

possible motions of three point

masses m1,m2,and m3, which

attract each other according to

Newton's law of inverse squares.

In 1977, Mitchell Feigenbaum published the noted

article “ Quantitative Universality for a Class of

Nonlinear Transformations", where he described

logistic maps. Feigenbaum notably discovered the

universality in chaos, permitting an application of

chaos theory to many different phenomena.

History …

Edward Lorenz whose interest in chaos

came about accidentally through his work

on weather prediction in 1961.

small changes in initial conditions

produced large changes in the long-term

outcome. Predictability: Does the Flap of

a Butterfly’s Wings in Brazil set off a

Tornado in Texas?

History …

The flapping of a single butterfly's wing today produces a tiny

change in the state of the atmosphere. Over a period of time,

what the atmosphere actually does diverges from what it would

have done. So, in a month's time, a tornado that would have

devastated the Indonesian coast doesn't happen. Or maybe one

that wasn't going to happen, does.

(Ian Stewart, Does God Play Dice? The Mathematics of Chaos,

pg. 141)

Butterfly Effect

The father of fractals: Gaston Julia. 1900

There were some other works out there, such as

Sierpinski’s triangle and Koch’s curve.

Mandelbrot 1970 :Mandelbrot Set.

History of Fractals

Fractals …

Koch’s curveFractals …

Fractals …

Fractals …

Types of Attractors:

Types of Attractors:

Self Similarity in Chaos

• mean• variance• power

spectrum

Chaos and stochastic process

Similar time series

RANDOMrandomx(n) = RND

CHAOSDeterministicx(n+1) = 3.95 x(n) [1-x(n)]

How to recognize chaos from random

Power spectraStructure in state spaceDimension of dynamicsSensitivity to initial condition Lyapunov ExponentsPredictive Ability

Controllability of Chaos

Structure in state space

Poincare Section

divergence

Divergence

Divergence

Divergence

Predictive Ability

)1(1 nnn xrxx

Logistic Map

0 < r < 3 The sequence approaches to a stable value.

3 < r < 3.570 The sequence jumps among some stable values.

3.570 < r < 4 The sequence shows chaotic behavior.

Feigenbaum Number

Biological models producing chaosNonlinearity

Time delay

Compartment Cascades

Forcing Functions

Chaos in Biology

why chaos is so important in Biology?

Chaotic systems can be used to show :

rhythms of heartbeats

walking strides

Fractals can be used to model:

Structures of nerve networks

circulatory systems

lungs

DNA

why chaos is so important in Biology?

Evidence for chaotic healthy hearts

Applying chaos theory to a

cardiac arrhythmia

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